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2,is used more in civic areas,like 电话号码（phone number） ,门牌号码（door number）,QQ号码（if you use qq），身份证号码（ID number），数字 is used more in mathematics（especially arithmetic）。3. 期盼 is almost completely synonymous for “look forward to”，for an oncoming event namely。希望 is more like “hope”or“wish”stressing you want sth in a certain way。

Well, the differences between the first pair mainly derive from the collocations in which they are used. When you say 'the number of a card', you say 卡片的号码，but people never say 卡片的数字。Similarly, we say 电话号码 instead of 电话数字.

There are cases where only 数字 is used: 你的幸运数字是什么？(What's your lucky number?)

As for the second pair, 期盼 is better translated as 'be looking forward to', while 希望 is better translated as 'hope'.